A combination of novel technological and fundamental physics prospects has sparked a huge interest in pure spin transport in magnets, starting with ferromagnets and spreading to antiferro- and ferrimagnets. We present a theoretical study of spin transport across a ferrimagnet-nonmagnetic conductor interface, when a magnetic eigenmode is driven into a coherent state. The obtained spin current expression includes intra- as well as cross-sublattice terms, both of which are essential for a quantitative understanding of spin pumping. The dc current is found to be sensitive to the asymmetry in interfacial coupling between the two sublattice magnetizations and the mobile electrons, especially for antiferromagnets. We further find that the concomitant shot noise provides a useful tool for probing the quasiparticle spin and interfacial coupling.